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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Uniform absolute continuity in spaces of set functions


Author: James D. Stein
Journal: Proc. Amer. Math. Soc. 51 (1975), 137-140
MSC: Primary 28A32
MathSciNet review: 0440012
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Abstract: Let $ X$ be a regular topological space, $ K$ a collection of bounded regular measures defined on the Borel sets of $ X$. The following conditions are equivalent.

(1) Let $ M(X)$ denote the Borel measures, $ M{(X)^ + }$ the nonnegative members of $ M(X)$. There is a $ \lambda \in M{(X)^ + }$ such that $ K$ is uniformly $ \lambda $-continuous.

(2) If $ \{ {U_n}\vert n = 1,2, \ldots \} $ is a disjoint sequence of open sets, then $ {\lim _{{n^{ \to \infty }}}}\mu ({U_n}) = 0$ uniformly for $ \mu \in K$.

(3) If $ E$ is a Borel subset of $ X$ and $ \epsilon > 0$, there is a compact set $ F \subseteq E$ such that $ \vert\mu \vert(E \sim F) < \epsilon $ for $ \mu \in K$.

(4) If $ \{ {E_n}\vert n = 1,2, \ldots \} $ is a disjoint sequence of Borel sets, then $ {\lim _{n \to \infty }}\mu ({E_n}) = 0$ uniformly for $ \mu \in K$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0440012-0
PII: S 0002-9939(1975)0440012-0
Article copyright: © Copyright 1975 American Mathematical Society